I found the following relation which answers my question
$$R=\log\det \left(\boldsymbol{C}_{MMSE}^{-1}\right)$$
where $\boldsymbol{C}_{MMSE}$ is the covariance matrix of the error $\boldsymbol{e}=\boldsymbol{h}-\hat{\boldsymbol{h}}$
I found the answer in the paper "An Iteratively Weighted MMSE Approach to Distributed Sum-Utility Maximization for a MIMO Interfering Broadcast Channel" but it is said to be well known